Understanding the Given Trigonometric Functions

Oct 13, 2025 by miodeya.com 79 views

Introduction

In this article, we will explore the process of sketching and labeling a right triangle using the given trigonometric functions $\sin \theta=\frac{3}{7}$ and $\cot \theta=\frac{2 \sqrt{10}}{3}$ . We will also evaluate the other four trigonometric functions, namely $\cos \theta$ , $\tan \theta$ , $\csc \theta$ , and $\sec \theta$ , to gain a deeper understanding of the relationships between the sides and angles of the triangle.

Understanding the Given Trigonometric Functions

The given trigonometric functions are $\sin \theta=\frac{3}{7}$ and $\cot \theta=\frac{2 \sqrt{10}}{3}$ . The sine function represents the ratio of the length of the opposite side to the length of the hypotenuse, while the cotangent function represents the ratio of the length of the adjacent side to the length of the opposite side.

Sine Function

The sine function is defined as $\sin \theta=\frac{\text{opposite}}{\text{hypotenuse}}$ . In this case, we are given that $\sin \theta=\frac{3}{7}$ . This means that the length of the opposite side is 3 units, and the length of the hypotenuse is 7 units.

Cotangent Function

The cotangent function is defined as $\cot \theta=\frac{\text{adjacent}}{\text{opposite}}$ . In this case, we are given that $\cot \theta=\frac{2 \sqrt{10}}{3}$ . This means that the length of the adjacent side is $2 \sqrt{10}$ units, and the length of the opposite side is 3 units.

Sketching and Labeling the Triangle

Using the given trigonometric functions, we can sketch and label the right triangle. The opposite side has a length of 3 units, and the hypotenuse has a length of 7 units. The adjacent side has a length of $2 \sqrt{10}$ units.

Labeling the Triangle

We can label the triangle as follows:

The opposite side is labeled as $3$ .
The adjacent side is labeled as $2 \sqrt{10}$ .
The hypotenuse is labeled as $7$ .

Evaluating the Other Four Trigonometric Functions

We can use the given trigonometric functions to evaluate the other four trigonometric functions, namely $\cos \theta$ , $\tan \theta$ , $\csc \theta$ , and $\sec \theta$ .

Cosine Function

The cosine function is defined as $\cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}}$ . We can use the given values to evaluate the cosine function:

\cos \theta=\frac{2 \sqrt{10}}{7}

Tangent Function

The tangent function is defined as $\tan \theta=\frac{\text{opposite}}{\text{adjacent}}$ . We can use the given values to evaluate the tangent function:

\tan \theta=\frac{3}{2 \sqrt{10}}

Cosecant Function

The cosecant function is defined as $\csc \theta=\frac{\text{hypotenuse}}{\text{opposite}}$ . We can use the given values to evaluate the cosecant function:

\csc \theta=\frac{7}{3}

Secant Function

The secant function is defined as $\sec \theta=\frac{\text{hypotenuse}}{\text{adjacent}}$ . We can use the given values to evaluate the secant function:

\sec \theta=\frac{7}{2 \sqrt{10}}

Conclusion

In this article, we have explored the process of sketching and labeling a right triangle using the given trigonometric functions $\sin \theta=\frac{3}{7}$ and $\cot \theta=\frac{2 \sqrt{10}}{3}$ . We have also evaluated the other four trigonometric functions, namely $\cos \theta$ , $\tan \theta$ , $\csc \theta$ , and $\sec \theta$ , to gain a deeper understanding of the relationships between the sides and angles of the triangle.

References

[1] "Trigonometry" by Michael Corral, 2015.
[2] "Precalculus: Mathematics for Calculus" by James Stewart, 2015.

Glossary

Sine: The ratio of the length of the opposite side to the length of the hypotenuse.
Cosine: The ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent: The ratio of the length of the opposite side to the length of the adjacent side.
Cosecant: The ratio of the length of the hypotenuse to the length of the opposite side.
Secant: The ratio of the length of the hypotenuse to the length of the adjacent side.

Introduction

In our previous article, we explored the process of sketching and labeling a right triangle using the given trigonometric functions $\sin \theta=\frac{3}{7}$ and $\cot \theta=\frac{2 \sqrt{10}}{3}$ . We also evaluated the other four trigonometric functions, namely $\cos \theta$ , $\tan \theta$ , $\csc \theta$ , and $\sec \theta$ , to gain a deeper understanding of the relationships between the sides and angles of the triangle. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the relationship between the sine and cosine functions?

A: The sine and cosine functions are related by the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ . This means that the sum of the squares of the sine and cosine functions is equal to 1.

Q: How do I sketch a right triangle using the given trigonometric functions?

A: To sketch a right triangle using the given trigonometric functions, you can use the following steps:

Draw a horizontal line to represent the adjacent side.
Draw a vertical line to represent the opposite side.
Use the given values to determine the length of the opposite side and the hypotenuse.
Use the Pythagorean theorem to determine the length of the adjacent side.

Q: What is the difference between the tangent and cotangent functions?

A: The tangent and cotangent functions are reciprocals of each other. The tangent function is defined as $\tan \theta=\frac{\text{opposite}}{\text{adjacent}}$ , while the cotangent function is defined as $\cot \theta=\frac{\text{adjacent}}{\text{opposite}}$ .

Q: How do I evaluate the cosecant and secant functions?

A: To evaluate the cosecant and secant functions, you can use the following formulas:

$\csc \theta=\frac{\text{hypotenuse}}{\text{opposite}}$
$\sec \theta=\frac{\text{hypotenuse}}{\text{adjacent}}$

Q: What is the relationship between the sine and cosecant functions?

A: The sine and cosecant functions are reciprocals of each other. The sine function is defined as $\sin \theta=\frac{\text{opposite}}{\text{hypotenuse}}$ , while the cosecant function is defined as $\csc \theta=\frac{\text{hypotenuse}}{\text{opposite}}$ .

Q: How do I use the given trigonometric functions to evaluate the other four trigonometric functions?

A: To use the given trigonometric functions to evaluate the other four trigonometric functions, you can use the following formulas:

$\cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan \theta=\frac{\text{opposite}}{\text{adjacent}}$
$\csc \theta=\frac{\text{hypotenuse}}{\text{opposite}}$
$\sec \theta=\frac{\text{hypotenuse}}{\text{adjacent}}$

Conclusion

In this article, we have answered some frequently asked questions related to the topic of sketching and labeling a right triangle using the given trigonometric functions. We hope that this article has been helpful in clarifying any confusion and providing a deeper understanding of the relationships between the sides and angles of the triangle.

References

[1] "Trigonometry" by Michael Corral, 2015.
[2] "Precalculus: Mathematics for Calculus" by James Stewart, 2015.

Glossary

Sine: The ratio of the length of the opposite side to the length of the hypotenuse.
Cosine: The ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent: The ratio of the length of the opposite side to the length of the adjacent side.
Cosecant: The ratio of the length of the hypotenuse to the length of the opposite side.
Secant: The ratio of the length of the hypotenuse to the length of the adjacent side.